Due to space limitation, an outline of the proof is given. First note
the sub-problem of \(\mathbf {W}\)  is convex. For the sub-problem of
\(\Theta _{l}\)  in (REF )
or (REF ) with nonconvex constraint \(\mathcal {R}_{l}\) ,
it can be verified that the regularity condition liner independence
constraint qualification (LICQ) [1]} holds everywhere
on set \(\mathcal {R}_{l}\) . The continuous phase constraint \(\mathcal {R}_{l}^{\mathsf {C}}\) 
and the discrete phase constraint \(\mathcal {R}_{l}^{\mathsf {D}}\) 
w.r.t. \(\theta _{l}\)  can be rewritten as equality constraints
\([\mathbf {h}_{l}^{\mathsf {C}}(\theta _{l})]_{i}=([\theta _{l}]_{i})^{2}-1=0\) ,
\(i=1,\ldots ,N_{l}\)  and \([\mathbf {h}_{l}^{\mathsf {D}}]_{i}(\theta _{l})=\prod _{p=1,\cdots ,P_{l}}\bigl ([\theta _{l}]_{i}-e^{j\phi _{lp}}\bigr )=0\) ,
\(i=1,\ldots ,N_{l}\) , respectively. The gradient \(\nabla [\mathbf {h}_{l}^{\mathsf {C}}(\theta _{l})]_{1},\ldots ,\nabla [\mathbf {h}_{l}^{\mathsf {C}}(\theta _{l})]_{N_{l}}\) 
and \(\nabla [\mathbf {h}_{l}^{\mathsf {D}}(\theta _{l})]_{1},\ldots ,\nabla [\mathbf {h}_{l}^{\mathsf {D}}(\theta _{l})]_{N_{l}}\) 
meet the LICQ evidently since \([\theta _{l}]_{j}\)  appears
only at the \(j\) -the entry of vector \(\nabla [\mathbf {h}_{l}^{\mathsf {C}}(\theta _{l})]_{i}\) 
and \(\nabla [\mathbf {h}_{l}^{\mathsf {D}}(\theta _{l})]_{i}\) .
Therefore, the updated \(\theta _{l}\)  via Lemma REF 
is a KKT point of problem (REF )
or problem (REF ). Considering the properties
of minorizing functions [2]}, it can be shown
\(\bigl \lbrace \!\lbrace \mathbf {w}_{i}^{(\infty )}\rbrace ,\!\lbrace \mathbf {\Theta }_{i}^{(\infty )}\rbrace \!\bigr \rbrace \!\) 
is a KKT point of Problem (REF ).
